angle between line and plane

The angle between a line l and a plane τ is defined as the least possible angle ω between l and a line contained by τ.

It is apparent that ω satisfies always 0≦ω≦90∘.

Let the plane τ be given by the equation (http://planetmath.org/EquationOfPlane) A⁢x+B⁢y+C⁢z+D=0, i.e. its normal vector has the componentsA,B,C. Let a direction vector of the line l have the components a,b,c. Then the angle ω between l and τ is obtained from the equation

Example. Consider the x⁢y-plane and the line l through the origin and the point (1, 1, 1). We can use the components 1, 1, 1 for the direction vector of l and the components 0, 0, 1 for the normal vector of the plane. We have